1. Technical Field
The present invention relates to a cut of a resonator making use of a piezoelectric effect and, more particularly, to a cut of a piezoelectric resonator, piezoelectric resonator, and piezoelectric device using a so-called new cut quartz plate.
2. Related Art
As various kinds of electronic devices have been advanced and communication systems have evolved in recent years, piezoelectric devices typified by piezoelectric resonators have been used frequently. Especially, quartz acting as a piezoelectric material has enjoyed wide acceptance in piezoelectric devices, because high frequencies are obtained and stable frequency characteristics are provided. AT-cut quartz plates (hereinafter simply abbreviated AT-cut plates) have been used in piezoelectric devices for a long time because piezoelectric resonators having stable frequency characteristics in a wide temperature range are obtained. Such an AT-cut plate has one side parallel to the X axis and has been cut at a cut angle obtained by rotating the XZ-plane with 35.25 degrees in a clockwise direction (as viewed from the −X direction of the X axis to the +X direction) about the X axis.
In recent years, however, as resonator and so on have been packed at increasing densities, the operating temperatures have been elevated. Also, it has become necessary to set the operating temperatures of the resonators higher. Therefore, a doubly-rotated resonator whose cut angles are rotated about two axes has been devised instead of the conventional AT-cut resonator.
If a quartz plate (doubly-rotated substrate) cut out with cut angles rotated relative to two axes among the crystallographic axes (electric axis, mechanical axis, and optic axis) of quartz is used, it has been theoretically demonstrated that the central temperature of the frequency-temperature characteristics shifts to the higher temperature side. In a temperature range of from −25° to +100° C., cut angles providing stable frequencies exist (for example, see Japanese Patent No. 3,218,537).
The doubly-rotated substrate can provide stable frequency-temperature characteristics in this way. At the same time, many spurious modes occur compared with the main mode. Many frequency jumps or resistance value increases occur due to mechanical oscillation coupling of spurious modes with the main mode. Many of the spurious modes are contour vibrations depending on the longer or shorter sides of blanks or are modes of combinations of them. Accordingly, when a blank is designed, its shape must be determined carefully such that no spurious modes exist near the frequency of the main mode.
The temperature characteristics of the doubley-rotated resonator are as shown in FIG. 3 when the frequency-temperature characteristics in the temperature range from −25° C. to +100° C. of the resonator are taken which has rotated with 34.9 degrees about the X axis after rotating 10 degrees about the Z axis, for example. In the figure, the dotted line indicate the frequency-temperature characteristics of the conventional AT-cut resonator. It can be seen that the doubly-rotated resonator shows more stable frequency-temperature characteristics in higher temperature regions as compared to the AT-cut resonator. In this doubly-rotated resonator, the frequency-temperature characteristics have a point at which the gradient of the tangential line is 0 near approximately 25° C. in the case of the AT-cut. The temperature at this point is hereinafter referred to as the central temperature. In contrast, the doubly-rotated resonator has the phenomenon that the central temperature varies from 25° C. to 100° C. or higher depending on the rotational angle φ.
The characteristics of the doubly-rotated resonator described so far are useful. However, more spurious modes (undesired modes) having other modes of vibration are produced than the conventional AT-cut resonator. For example, contour vibrations depending on the longer and shorter sides of the blank vary in frequency due to deviation in blank contour. Therefore, if the frequency comes too close to that of the main modes, both vibrations are mechanically coupled, causing jumps in the frequency of the main modes or resistance increases. Similarly, in the temperature range of from −25° C. to 100° C. taken as operating temperatures, spurious modes occur at certain temperatures near the main modes. This produces the phenomenon that the frequency of the main modes deviates or the resistance value increases. Similar phenomena are observed with AT-cut resonators. Especially, in the case of double rotations, these phenomena occur frequently. The present inventors have investigated the cause using an analytical method known as the finite element method.
As a result, we have found that the cause is the direction of displacement of thickness shear mode that is the main mode. The conventional AT-cut resonator produces a thickness shear mode whose direction of displacement is only in the X-direction. On the other hand, the direction of displacement of the doubly-rotated resonator has all of the components of X, Y, and Z. The phenomenon of a frequency shift or resistance increase caused by spurious mode is due to the fact that the main and spurious mode have common displacement components and thus cause coupling of vibrations (resonance). That is, the main modes of the AT-cut resonator couple with vibration having only X-direction displacement components. However, the doubly-rotated resonator has displacement components of three directions and so there arises the possibility that coupling with the majority of spurious modes occurs.